Minimum mean absolute error estimation over the class of generalized stack filters

A class of sliding window operators called generalized stack filters is developed. This class of filters, which includes all rank order filters, stack filters, and digital morphological filters, is the set of all filters possessing the threshold decomposition architecture and a consistency property called the stacking property. Conditions under which these filters possess the weak superposition property known as threshold decomposition are determined. An algorithm is provided for determining a generalized stack filter which minimizes the mean absolute error (MAE) between the output of the filter and a desired input signal, given noisy observations of that signal. The algorithm is a linear program whose complexity depends on the window width of the filter and the number of threshold levels observed by each of the filters in the superposition architecture. The results show that choosing the generalized stack filter which minimizes the MAE is equivalent to massively parallel threshold-crossing decisions making when the decision are consistent with each other